Soit E une courbe elliptique définie sur un corps de nombres K, et soit S un ensemble de densité 1 de places de K en lesquelles E a bonne réduction. Faltings a montré en 1983 que la classe de K-isogénie de E est caracterisée par la fonction , qui envoie chaque place sur lʼordre du groupe des points de E sur le corps résiduel correspondant. On montre quʼil suffit de considérer les nombres premiers divisant cet ordre.
Let E be an elliptic curve defined over a number field K and let S be a density-one set of primes of K of good reduction for E. Faltings proved in 1983 that the K-isogeny class of E is characterized by the function , which maps a prime to the order of the group of points of E over the corresponding field . We show that, in this statement, the integer can be replaced by its radical.
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@article{CRMATH_2013__351_1-2_1_0, author = {Hall, Chris and Perucca, Antonella}, title = {On the prime divisors of the number of points on an elliptic curve}, journal = {Comptes Rendus. Math\'ematique}, pages = {1--3}, publisher = {Elsevier}, volume = {351}, number = {1-2}, year = {2013}, doi = {10.1016/j.crma.2013.01.003}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2013.01.003/} }
TY - JOUR AU - Hall, Chris AU - Perucca, Antonella TI - On the prime divisors of the number of points on an elliptic curve JO - Comptes Rendus. Mathématique PY - 2013 SP - 1 EP - 3 VL - 351 IS - 1-2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2013.01.003/ DO - 10.1016/j.crma.2013.01.003 LA - en ID - CRMATH_2013__351_1-2_1_0 ER -
%0 Journal Article %A Hall, Chris %A Perucca, Antonella %T On the prime divisors of the number of points on an elliptic curve %J Comptes Rendus. Mathématique %D 2013 %P 1-3 %V 351 %N 1-2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2013.01.003/ %R 10.1016/j.crma.2013.01.003 %G en %F CRMATH_2013__351_1-2_1_0
Hall, Chris; Perucca, Antonella. On the prime divisors of the number of points on an elliptic curve. Comptes Rendus. Mathématique, Tome 351 (2013) no. 1-2, pp. 1-3. doi : 10.1016/j.crma.2013.01.003. http://archive.numdam.org/articles/10.1016/j.crma.2013.01.003/
[1] Finiteness theorems for Abelian varieties over number fields (Cornell, G.; Silverman, J.H., eds.), Arithmetic Geometry, Springer-Verlag, New York, 1986, pp. 9-27
[2] Horizontal isogeny theorems, Forum Math., Volume 14 (2002) no. 6, pp. 931-952
[3] Radical characterizations of elliptic curves, 2011 (preprint) | arXiv
[4] Points de torsion sur les variétés abéliennes de type GSp, J. Inst. Math. Jussieu, Volume 11 (2012) no. 1, pp. 27-65
[5] Elliptic Functions, Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987
[6] Isogénies horizontales et classes dʼisogénies de variétés abéliennes, 2012 (preprint) | arXiv
[7] Propriétés galoisiennes des points dʼordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972) no. 4, pp. 259-331
[8] Abelian varieties over finite fields, Ann. Sci. École Norm. Sup., Volume 4 (1969) no. 2, pp. 521-560
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